The fusion of rhythm, beat perception, and movement is often summarized under the term "entrainment" and becomes obvious when we effortlessly tap our feet or snap our fingers to the pulse of music. Entrainment to music involves a large network of brain structures, and neural oscillations at beat-related frequencies can help elucidate how this network is connected. Here, we used EEG to investigate steady-state evoked potentials (SSEPs) and event-related potentials (ERPs) during listening and tapping to drum clips with different rhythmic structures that were interrupted by silent breaks of 2-6 sec. This design allowed us to address the question of whether neural entrainment processes persist after the physical presence of musical rhythms and to link neural oscillations and event-related neural responses. During stimulus presentation, SSEPs were elicited in both tasks (listening and tapping). During silent breaks, SSEPs were only present in the tapping task. Notably, the amplitude of the N1 ERP component was more negative after longer silent breaks, and both N1 and SSEP results indicate that neural entrainment was increased when listening to drum rhythms compared with an isochronous metronome. Taken together, this suggests that neural entrainment to music is not solely driven by the physical input but involves endogenous timing processes. Our findings break ground for a tighter linkage between steady-state and transient evoked neural responses in rhythm processing. Beyond music perception, they further support the crucial role of entrained oscillatory activity in shaping sensory, motor, and cognitive processes in general.